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Error Stability Properties of Generalized Gradient-Type Algorithms

Author

Listed:
  • M. V. Solodov

    (Instituto de Matemática Pura e Aplicada)

  • S. K. Zavriev

    (Moscow State University)

Abstract

We present a unified framework for convergence analysis of generalized subgradient-type algorithms in the presence of perturbations. A principal novel feature of our analysis is that perturbations need not tend to zero in the limit. It is established that the iterates of the algorithms are attracted, in a certain sense, to an ɛ-stationary set of the problem, where ɛ depends on the magnitude of perturbations. Characterization of the attraction sets is given in the general (nonsmooth and nonconvex) case. The results are further strengthened for convex, weakly sharp, and strongly convex problems. Our analysis extends and unifies previously known results on convergence and stability properties of gradient and subgradient methods, including their incremental, parallel, and heavy ball modifications.

Suggested Citation

  • M. V. Solodov & S. K. Zavriev, 1998. "Error Stability Properties of Generalized Gradient-Type Algorithms," Journal of Optimization Theory and Applications, Springer, vol. 98(3), pages 663-680, September.
    • Handle: RePEc:spr:joptap:v:98:y:1998:i:3:d:10.1023_a:1022680114518
    DOI: 10.1023/A:1022680114518
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    References listed on IDEAS

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    1. O. L. Mangasarian, 1993. "Mathematical Programming in Neural Networks," INFORMS Journal on Computing, INFORMS, vol. 5(4), pages 349-360, November.
    2. M. V. Solodov, 1997. "Convergence Analysis of Perturbed Feasible Descent Methods," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 337-353, May.
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    Cited by:

    1. Regina S. Burachik & Yaohua Hu & Xiaoqi Yang, 2022. "Interior quasi-subgradient method with non-Euclidean distances for constrained quasi-convex optimization problems in hilbert spaces," Journal of Global Optimization, Springer, vol. 83(2), pages 249-271, June.
    2. Matthias Rottmann & Kira Maag & Mathis Peyron & Hanno Gottschalk & Nataša Krejić, 2023. "Detection of Iterative Adversarial Attacks via Counter Attack," Journal of Optimization Theory and Applications, Springer, vol. 198(3), pages 892-929, September.
    3. M. V. Solodov, 2003. "On Approximations with Finite Precision in Bundle Methods for Nonsmooth Optimization," Journal of Optimization Theory and Applications, Springer, vol. 119(1), pages 151-165, October.
    4. Xiaojing Xu & Jinpeng Ma & Xiaoping Xie, 2019. "Price Convergence under a Probabilistic Double Auction," Computational Economics, Springer;Society for Computational Economics, vol. 54(3), pages 1113-1155, October.
    5. Wenma Jin & Yair Censor & Ming Jiang, 2016. "Bounded perturbation resilience of projected scaled gradient methods," Computational Optimization and Applications, Springer, vol. 63(2), pages 365-392, March.
    6. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 2003. "On the convergence of conditional [var epsilon]-subgradient methods for convex programs and convex-concave saddle-point problems," European Journal of Operational Research, Elsevier, vol. 151(3), pages 461-473, December.
    7. Elena Tovbis & Vladimir Krutikov & Predrag Stanimirović & Vladimir Meshechkin & Aleksey Popov & Lev Kazakovtsev, 2023. "A Family of Multi-Step Subgradient Minimization Methods," Mathematics, MDPI, vol. 11(10), pages 1-24, May.
    8. Peng Zhang & Gejun Bao, 2018. "An Incremental Subgradient Method on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 711-727, March.
    9. S. Sundhar Ram & A. Nedić & V. V. Veeravalli, 2010. "Distributed Stochastic Subgradient Projection Algorithms for Convex Optimization," Journal of Optimization Theory and Applications, Springer, vol. 147(3), pages 516-545, December.
    10. Jinpeng Ma & Qiongling Li, 2016. "Convergence of price processes under two dynamic double auctions," The Journal of Mechanism and Institution Design, Society for the Promotion of Mechanism and Institution Design, University of York, vol. 1(1), pages 1-44, December.
    11. Grégory Emiel & Claudia Sagastizábal, 2010. "Incremental-like bundle methods with application to energy planning," Computational Optimization and Applications, Springer, vol. 46(2), pages 305-332, June.
    12. Xiaoliang Wang & Liping Pang & Qi Wu & Mingkun Zhang, 2021. "An Adaptive Proximal Bundle Method with Inexact Oracles for a Class of Nonconvex and Nonsmooth Composite Optimization," Mathematics, MDPI, vol. 9(8), pages 1-27, April.

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